模态截断对悬索非线性耦合共振响应影响

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振动与冲击 ›› 2023, Vol. 42 ›› Issue (3) : 50-56.

论文

模态截断对悬索非线性耦合共振响应影响

张昕涛1，赵珧冰1， 2，蔡绍辉1，郭智锐1

作者信息 +

Effects of mode truncation on nonlinear coupled resonant responses of suspended cable

ZHANG Xintao1, ZHAO Yaobing1,2, CAI Shaohui1, GUO Zhirui1

Author information +

文章历史 +

摘要

对于各类动力系统共振响应，可以采用直接法和离散法得到其微分方程的近似解，而解的误差取决于两方面：模态离散和摄动分析。其中离散法采用有限模态来描述连续系统的动力学行为，如果忽略高阶模态振型和频率，定会带来一定误差，甚至无法反映真实的非线性动力学现象。因此无论是工程实践还是理论分析，离散法中模态截断带来的误差和收敛性备受关注。以水平悬索两正对称模态之间发生耦合共振为例，探究两种模态截断对该系统共振响应影响。首先利用Galerkin法得到离散后的面内运动微分方程，然后采用多尺度法求得系统发生耦合共振时的调制方程。通过对比激励响应幅值曲线、幅频响应曲线、时程曲线、相位图、频率谱、庞加莱截面和李雅普诺夫指数等，定量和定性地展示两阶和九阶模态截断导致的系统动力学行为差异。研究结果表明：非直接激励模态和非内共振模态对系统内共振响应存在影响，根源在于平方非线性的共振项；对于外激励直接作用于低阶和高阶模态的情况，由于模态截断导致的振动特性差异程度，前者要明显高于后者；在大幅共振区域，模态截断对系统响应幅值影响较为明显；分岔现象与模态截断阶数关系密切，倘若仅考虑两阶模态，结果可能会遗漏鞍结点分岔或出现额外的霍普夫分岔，从而导致跳跃现象和动态周期解发生明显改变；不同阶模态截断可能导致动力系统吸引子类型截然不同。

Abstract

As to many kinds of dynamical systems’ resonant responses, the direct and discretized methods could be adopted to obtain the approximate solutions. These solutions’ errors are dependent on two aspects: mode discretization and perturbation analysis. Using finite modes to describe the dynamical behaviors of the continuous systems will induce some errors, and the higher-order mode shapes and natural frequencies are neglected, leading to distortion of nonlinear dynamic phenomena. Therefore, no matter in the practical engineering or theoretical analysis, the errors and convergence of modal truncations in discretized method should be paid much attention. Here, based on the internal resonances between two symmetric modes of horizontal suspended cables, the influences of two different mode truncations on system’s resonant responses are investigated. Firstly, the discretized planar nonlinear vibration equations of motions are obtained by using the Galerkin method. Then, the modulation equations are obtained by using the multiple scales method. By comparing the force/frequency response amplitude curves, time history curves, phase plane diagrams, Poincare sections, and Lyapunov exponents, the fluences of two and nine mode truncations on the system’s dynamical behaviors are illustrated in detail. The numerical results show that: as to the internal resonances, the non-direct excited and non-internal resonant modes would affect the resonant responses definitely, and the cause lies in the resonant terms induced by the quadratic nonlinearity. The external excitation is applied on the lower or higher-order mode, and as to the differences induced by mode truncation, the former is significantly higher than the latter. The influences of mode truncations on the response amplitudes seem more obvious in the large resonant regions. The bifurcations are closely related to the mode truncation, and some saddle-node bifurcations might be missed and some extra Hopf bifurcations may be found when two modes discretization is adopted in the case. In these circumstances, the jump phenomena and the dynamic periodic solutions are changed significantly. Different orders of mode truncation could lead to very different systems’ attractors.

导出引用

张昕涛1，赵珧冰1， 2，蔡绍辉1，郭智锐1. 模态截断对悬索非线性耦合共振响应影响[J]. 振动与冲击, 2023, 42(3): 50-56

ZHANG Xintao1, ZHAO Yaobing1,2, CAI Shaohui1, GUO Zhirui1. Effects of mode truncation on nonlinear coupled resonant responses of suspended cable[J]. Journal of Vibration and Shock, 2023, 42(3): 50-56

参考文献

[1] Nayfeh A H. Introduction to perturbation techniques [M]. New York: Wiley, 1993.
[2] 陈立群. 关于对模态概念的理解[J]. 力学与实践, 2021, 43(2): 252-255.
CHEN Liqun. On the concept of modes[J]. Mechanics in Engineering, 2021, 43(2): 252-255.
[3] 孙东阳, 陈国平. 刚柔耦合多体系统动力学模型降阶[J]. 振动工程学报, 2014, 27(5): 708-714.
SUN Dongyang, CHEN Guoping. Model reduction of dynamics for rigid-flexible multi-systems[J]. Journal of Vibration Engineering, 2014, 27(5): 708-714.
[4] 王东, 张周锁. 连接局部迟滞非线性的时频域动力学降阶方法[J]. 振动工程学报, 2021, 34(3): 559-566.
WANG Dong, ZHANG Zhousuo. Dynamic reduction analysis for local hysteresis nonlinearity of joint interfaces in time and frequency domains[J]. Journal of Vibration Engineering, 2021, 34(3): 559-566.
[5] Rega G, Lacarbonara W, Nayfeh A H, et al. Multiple resonances in suspended cables: direct versus reduced-order models[J]. International Journal of Non-linear Mechanics, 1999, 34(5): 901-924.
[6] 金栋平, 胡海岩. 结构碰撞振动的建模与模态截断[J]. 固体力学学报, 2001, 22(2): 205-208.
JIN Dongping, HU Haiyan. Modeling and mode truncation of vibro-impacting structures[J]. Acta Mechanica Solida Sinica, 2001, 22(2): 205-208.
[7] 陈晓明, 冯志华, 张风君, 等. 含集中质量矩形薄板的动力学建模与质量调幅分析[J]. 应用力学学报, 2018, 35(4): 743-749+929.
CHEN Xiaoming, FENG Zhihua, ZHANG Fengjun, et al. Dynamic modeling of rectangular plates with a lumped mass and the amplitude modification effect of mass[J]. Chinese Journal of Applied Mechanics, 2018, 35(4): 743-749+929.
[8] 孙芳锦, 祝东涵, 张大明. 风力机叶片流固耦合计算的降阶模型研究[J]. 振动与冲击, 2021, 40(15): 175-181+215.
SUN Fangjin, ZHU Donghan, ZHANG Daming. Reduced order model for fluid-structure coupled calculation of wind turbine blades[J]. Journal of Vibration and Shock, 2021, 40(15): 175-181+215.
[9] 王乙坤, 王琳. 分布式运动约束下悬臂输液管的参数共振研究[J]. 力学学报, 2019, 51(2): 558-568.
WAND Yikun, WANG Lin. Parametric resonance of a cantilevered pipe conveying fluid subjected to distributed motion constraints[J]. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(2): 558-568.
[10] 郭勇, 谢建华. 微尺度悬臂管颤振的有限维研究[J]. 应用数学和力学, 2018, 39(2): 199-214.
GUO Yong, XIE Jianhua. Research on the flutter of micro-scale cantilever pipes-a finite-dimensional analysis[J]. Applied Mathematics and Mechanics, 2018, 39(2): 199-214.
[11] 齐欢欢, 徐鉴. Galerkin模态截断对计算悬臂输液管道固有频率的影响[J]. 振动与冲击, 2011, 30(1): 148-151.
QI Huanhuan, XU Jian. Effect of Galerkin modal truncation on natural frequency analysis of a cantilevered pipe conveying fluid[J]. Journal of Vibration and Shock, 2011, 30(1): 148-151.
[12] 熊柳杨, 张国策, 丁虎, 等. 黏弹性屈曲梁非线性内共振稳态周期响应[J]. 应用数学和力学, 2014, 35(11): 1188-1196.
XIONG Liuyang, ZHANG Guoce, DING Hu, et al. Steady-state periodic responses of a viscoelastic buckled beam in nonlinear internal resonance[J]. Applied Mathematics and Mechanics, 2014, 35(11): 1188-1196.
[13] Arafat H N, Nayfeh A H. Non-linear responses of suspended cables to primary resonance excitations[J]. Journal of Sound and Vibration, 2003, 266(2): 325-354.
[14] Guo T D, Rega G. Direct and discretized perturbations revisited: A new error source interpretation, with application to moving boundary problem[J]. European Journal of Mechanics/A Solids, 2020, 81: 103936.
[15] Guo T D, Rega G, Kang H. General perturbation correction: full-decomposition and physics-based elimination of non-secular terms[J]. International Journal of Mechanical Sciences, 2022, 216: 106966
[16] Luongo A, Zulli D, Piccardo G. Analytical and numerical approaches to nonlinear galloping of internally resonant suspended cables[J]. Journal of Sound and Vibration, 2008, 315(3): 375-393.
[17] Srinil N, Rega G. The effects of kinematic condensation on internally resonant forced vibrations of shallow horizontal cables[J]. International Journal of Non-Linear Mechanics, 2007, 42(1): 180-195.
[18] Zhao Y B, Sun C S, Wang Z Q, et al. Analytical solutions for resonant response of suspended cables subjected to external excitation[J]. Nonlinear Dynamics, 2014, 78(2): 1017-1032.
[19] Lacarbonara W, Rega G. Nayfeh A H. Resonant non-linear normal modes. Part I: analytical treatment for structural one-dimensional systems[J]. International. Journal of Non-linear Mechanics, 2003, 38(6): 851-872.